A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,
wherein each coefficient in the expansion of either A or B is real and
positive, and not less than the absolute value of the corresponding
coefficient in the expansion of a or b. In the second equation let us
substitute a series
F = P0 + tP1 + t^2P2/2! + ...,
wherein the coefficients in P0 are real and positive, and each not
less than the absolute value of the corresponding coefficient in p0;
then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary
equations of the same form as before, namely,
P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...
and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
[Delta]_s P0. These give for every coefficient in Ps+1 an integral
aggregate with real positive coefficients of the coefficients in P_s,
P_s-1, ..., P0 and the coefficients in A and B; and they are the same
aggregates as would be given by the previously obtained equations for
the corresponding coefficients in p_s+1 in terms of the coefficients
in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
coefficients in P0 and also in A, B are real and positive, it follows
that the values obtained in succession for the coefficients in P1, P2,
... are real and positive; and further, taking account of the fact
that the absolute value of a sum of terms is not greater than the sum
of the absolute values of the terms, it follows, for each value of s,
that every coefficient in p_s+1 is, in absolute value, not greater
than the corresponding coefficient in P_s+1. Thus if the series for F
be convergent, the series for f will also be; and we are thus reduced
to (1), specifying functions A, B with real positive coefficients,
each in absolute value not less than the corresponding coefficient in
a, b; (2) proving that the equation